Critical Point of Fluid Confined in Nanopores: Experimental Detection

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Critical Point of Fluid Confined in Nanopores: Experimental Detection and Measurement Sugata P. Tan, Xingdong Qiu, Morteza Dejam, and Hertanto Adidharma J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b00299 • Publication Date (Web): 28 Mar 2019 Downloaded from http://pubs.acs.org on March 28, 2019

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The Journal of Physical Chemistry

Critical Point of Fluid Confined in Nanopores: Experimental Detection and Measurement Sugata P. Tan,*,‡ Xingdong Qiu,† Morteza Dejam,† Hertanto Adidharma† ‡

†

Planetary Science Institute, Tucson, AZ 85719-2395, U.S.A.

Department of Petroleum Engineering, University of Wyoming, Laramie, WY 82071-2000, U.S.A.

ABSTRACT: For the first time, the presence of critical point of fluids confined in nanopores is unambiguously demonstrated and measured using Differential Scanning Calorimeter (DSC) with an isochoric cooling procedure. The behavior of critical point of CO 2 and C 2 H 6 in SBA-15 is found to deviate from theoretical predictions. The heat involved in capillary condensation is also discussed. The new and simple method of measurement introduced in this study opens the way to further investigate the criticality of confined fluids in nanopores.

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1. INTRODUCTION While it has been experimentally found that fluid condensation in confined space, such as in nanopores, isothermally occurs at lower pressures or isobarically at higher temperatures than that of free fluids in the bulk,1,2 the corresponding critical point has not been convincingly demonstrated and measured despite strong hints from theoretical work that it should exist at a lower temperature and pressure relative to its bulk counterpart.3,4 Beyond this point, analogous to the critical point of fluids in the bulk, condensation of confined fluids can no longer occur. The point is known as the pore critical point (PCP). Note that PCP is different from the hysteresis critical point (HCP), which is commonly derived from experimental adsorption-desorption isochores5 or isotherms.6 As widely known, adsorption isotherms at sufficiently low temperatures are not reversible. The adsorption path upon pressurizing is different from that of desorption upon depressurizing. This irreversible part of isotherms is known as hysteresis, which exists up to a relatively moderate temperature and pressure known as HCP. Beyond HCP, the isotherms become reversible, where the paths of adsorption and desorption are indistinguishable. HCP was thought to be the same as PCP, as clearly stated in an earlier work with sulfur hexafluoride on controlled-pore glass.7 Random porous mediums such as aerogels and glasses were popularly used in the early years. The measurement methods varied from sorption volumetric technique,5 specific heat,8 light scattering,9 ultrasonic method,10 to smallangle neutron scattering.11 Due to the weak fluid-wall interaction and large pore sizes of these mediums, however, the critical-temperature shift from bulk critical point (CP) was small and it was also impossible to reach equilibrium above HCP,12 the situation of which did not allow to clearly distinguish PCP from HCP. In all cases, the measured HCP’s were found to be at lower temperature and pressure relative to CP.


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Morishige and Shikimi were the first to provide evidence that HCP and PCP of a confined fluid are different points.13 PCP lies between HCP and the bulk critical point CP. As the slope of an adsorption isotherm at capillary condensation is expected to change at PCP upon temperature increase, the researchers also devised a graphical method to estimate the location of PCP. However, the slope change turned out to be gradual and so subtle that it was not easy to pinpoint the location. Recently, this slope method was applied to investigate the criticality of propane, nbutane, and n-pentane in MCM-41,14 where the PCP’s obtained were also rough estimates due to the same reason. There has not been any other experimental method to measure PCP. All these facts have made PCP good in concept but not in physical sense. Consequently, many researchers who work on confined-fluid phase equilibrium modeling just ignore PCP such that confined fluids reach criticality at the bulk CP.15-17 Before it can be shown experimentally that PCP does exist and can be measured with confidence, it is hard to convince them to include the pore criticality in their models. On the opposite side, some researchers did account for the effects of PCP in their equation of state (EOS),18,19 e.g., by applying prediction from theory,4 or using the PCP properties derived from molecular simulation,20 as their EOS parameters. These approaches undoubtedly suffer from the inaccuracy due to the assumptions made in the theory and simulation. Therefore, the only way that can lead to solving this difficult situation is to come up with a simple but reliable method to physically show the presence of PCP and experimentally measure it at reasonable accuracy. Recently, we developed a new method to measure the capillary condensation of pure fluids and mixtures confined in nanopores. It employs a micro DSC with isochoric cooling procedures that accurately measure the condensation.21,22 We will apply the


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procedure in this work to detect and measure the PCP of CO 2 and C 2 H 6 in SBA-15 with three different pore sizes. 2. EXPERIMENTAL METHOD AND RESULTS The experimental setup and materials used in this work were described in our recent publications.21,22 For the porous medium used in the experiments, i.e., SBA-15, we amend the pore sizes with the values determined using non-local density functional theory as recommended by IUPAC:23 4.570 nm, 6.079 nm, and 8.145 nm for samples S1, S2, and S3, respectively.

4.7

Heat flow [mW]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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supercritical

4.6

subcritical 4.5

peak

4.4 304

300

296

292

288

Temperature [K]

Figure 1. Typical thermograms from DSC along isochoric cooling paths at subcritical and supercritical conditions of the confined fluid. The original thermograms use time as the horizontal axis that also translates to simultaneously measured pressure or temperature. Typical thermograms along isochoric cooling paths are shown in Figure 1. At subcritical condition of the confined fluid, the path crosses the saturation curve where capillary condensation occurs, i.e. when the thermogram shows a distinct peak (at about 293 K in Figure 1). This peak


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becomes smaller and smaller as the condition approaches PCP, beyond which the capillary condensation no longer occurs; at this supercritical condition, the thermogram has no peak as also shown in Figure 1. The peak vanishes at PCP. The presence of the condensation peak may be represented using the area under the peak on the thermogram given by the DSC. Since the thermograms in Figure 1 originally use time as the horizontal axis corresponding to the temperature, the area under the peak is essentially the total heat involved in the capillary condensation. To quantitatively relate the peak area to the heat of capillary condensation, i.e. the heat per mole-fluid released at condensation, the amount of adsorbed fluid must be accurately measured, which is not easy to do. Fortunately, we just need the total heat to detect and investigate PCP in this work. As disclosed in our previous work22 for pure confined fluids, the capillary condensation measured using different amounts of adsorbent lies on the same saturation curve on the phase diagram. However, larger amounts of adsorbent, which allow more fluids in the confinement, lead to higher peaks on the thermograms for easy detection and accurate measurement. This fact turns out to be very useful to locate PCP as described next. Measurements of capillary condensation with a single weight of adsorbent cannot reach PCP because the heat peak can be so small near PCP that likely introduces inaccuracy. Therefore, three different weights of adsorbent were used for each type of fluid and adsorbent. The experimental data for fluids confined in S1, S2, and S3 are provided in the supporting information (SI). The larger weight enables measurement closer to PCP. As shown in Figure 2 for S1, we found that the total heat h is linear with the reciprocal of absolute temperature (1/T) and the logarithm of condensation pressure ln(P), even in the region where h almost vanishes. This behavior is different from that in the bulk, which follows the asymptotic power law near critical point. The difference


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is justifiable, however, considering that the density fluctuation near PCP is severely suppressed due to the confinement.24 260

5.5 mg

54.6

15 mg

In SBA-15 (S1)

In SBA-15 (S1)

CO2

46.0

270 30 mg

CO2

15 mg

45 mg

280

290 C2H6 300

Pressure [bar]: natural logarithmic scale

10 mg

Temperature [K]: reciprocal scale

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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37.3

28.7

15 mg 45 mg

C2H6 10 mg

20.1 5.5 mg

30 mg

15 mg

16.9

13.7

0.0

0.1

0.2

0.3

0.4

0.0

Heat [J] (a)

0.1

0.2

0.3

0.4

Heat [J] (b)

Figure 2. Three-line approach to obtain (a) T Cp and (b) P Cp in S1. Symbols: capillarycondensation data of CO 2 (open) and C 2 H 6 (filled); lines: regressions; stars: PCP’s. All lines in Figure 2 converge at the pore critical temperature (T Cp ) or pressure (P Cp ), the values of which will be experimentally verified in later discussion. For each fluid and adsorbent, T Cp and P Cp were determined such that the linear regressions of the three-line set resulted in the largest R2 coefficient. Table 1 lists the critical properties measured in this study. Therefore, in general, lines in a series of measurement for the same confined fluid and nanopores intersect the vertical axis at the same critical properties but with different slopes depending on the amount of adsorbent: b 1 1 = + T h T TCp vT ln P = ln PCp +

(1a)

bP h vP

(1b)


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where (b T /v T ) and (b P /v P ) are the slopes obtained from linear regression. The value of v may be set to unity for the smallest amount of adsorbent. As shown in Table 2, v T is virtually the same as v P for the same fluid and amount of adsorbent (w). Moreover, v is also similar to the ratio of adsorbent weight w for all lines, which indicates that v is related to the total pore volume available for the confined fluid. Cases for S2 and S3 are included in the SI. Tabel 1. Measured PCP of CO 2 and C 2 H 6 in SBA-15

CO 2

Bulk CP

T Cp [K], P Cp [bar] in SBA-15

T C [K], P C [bar]

S1

S2

S3

304.13, 73.77

294.5±0.4, 301.3±0.3, 303.0±0.2, 51.4±0.6 61.9±0.4 66.4±0.3

C 2 H 6 305.33, 48.72

297.5±0.4, 304.1±0.4, 304.6±0.5, 34.8±0.4 41.0±0.4 43.2±0.4

Tabel 2. The slopes of lines regressed to determine T Cp and P Cp for CO 2 and C 2 H 6 in S1 Linear function

w *w i /w 1 [mg]

vT

bT

vP

bP

5.5

1

1

10

1.82

1.987

15

2.73

3.644 1.983 -7.936 × 10-3 3.569 3.562

15

1

1

C 2 H 6 30

2.00

1.803

45

3.00

CO 2

1

1

2.732 1.807 -5.520 × 10-3 3.933 3.834

* Subscript 1 refers to the smallest w, while subscripts i refers to the other weights. For confirmation purposes, additional experimental tests were performed, as shown in Figure 3. As mentioned earlier, the heat peak on the thermogram vanishes at PCP. Measurements to generate


7


two cooling paths for fluids confined in S1 were taken, which are about 3-4 bar apart: one is a little above the measured PCP (supercritical path) and the other is well below PCP (subcritical path). The corresponding thermograms are also presented, which demonstrate that the heat peak appears on the subcritical path but not on the supercritical path, thus confirming PCP to lie somewhere in between. To the best our knowledge, this is for the first time where the presence and the location of PCP are unambiguously demonstrated and measured in experiments.

80

4.7

53.4

53.6

Pressure [bar] 53.2 53.0

52.8

Bulk saturated pressure (NIST) Supercritical cooling path Subcritical cooling path Bulk condensation

60

Pressure [bar]

Heat Flow [mW]

CO2 in SBA-15 (S1)

Supercritical

4.6

4.5

Capillary condensation 4.4

PCP

300

296

292

Temperature [K]

40

Pressure [bar] 49.7

20 260

270

280

290

300

Heat Flow [mW]

4.7

49.5

49.3

49.1

48.9

Subcritical

4.6

4.5

Temperature [K] 4.4 300

(a)

292

288

Pressure [bar]

60

35.3

C2H6 in SBA-15 (S1)

Capillary condensation

Heat Flow [mW]

40

35.2

35.0 34.9

35.1

4.6

Bulk saturated pressure (NIST) Supercritical cooling path Subcritical cooling path Bulk condensation

50

Pressure [bar]

296

Temperature [K]

Supercritical

4.5

4.4

PCP

304

300

296

292

Temperature [K]

30

Pressure [bar] 32.1

32.3

20

31.7

31.9

10 260

270

280

290

Temperature [K]

300

Heat Flow [mW]

4.6

Subcritical

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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4.5

4.4 304

(b)

300

296

292

Temperature [K]

288

Figure 3. Capillary condensation of: (a) CO 2 and (b) C 2 H 6 in S1 on the phase diagrams up to PCP. Symbols are experimental data corresponding to those in Figure 2. Thermograms of the cooling paths are shown on the right-hand side.


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3. DISCUSSION As compared to CP, PCP is indeed below CP for all cases. PCP for fluids in S1, which has the smallest pore size, is the lowest, consistent with their capillary condensation. However, the degree of decrease in temperature differs by one order of magnitude from that in pressure. For example, compared to bulk CP, T Cp in S1 is lower by 3%, while it is by 30% for P Cp (see Table 1). This is an immediate challenge to a theory based on van der Waals mean-field model for Lennard-Jones (LJ) fluids inside a cylindrical nanopore,4 which predicts the same behavior of T Cp and P Cp :

σ  = 1− = c1 − c2   1−  rp  TC PC rp   TCp

PCp

σ

2

(2)

where σ and r p are the LJ-molecular diameter and the pore radius, respectively. Obviously, the critical-point shifts vanish in the bulk free space, i.e. when r p goes to infinity. The coefficients c 1 and c 2 were derived to be 0.9409 and 0.2415, respectively.4 Another theoretical work, based on the density functional theory for fluids confined in cylindrical nanopores, predicted cubic behavior of the critical-temperature shift:3

σ  1− = k1 − k3    rp  TC rp   TCp

σ

3

(3)

where the coefficients k 3 = 0.375 k 1 , while k 1 is a constant related to the decay length of both fluidwall and fluid-fluid molecular interactions and commonly set to unity. As seen in Figure 4, eqs 2 and 3 are nearly linear and similar to each other. The values of σ used in Figure 4 are 0.407 nm (CO 2 ) and 0.3954 nm (C 2 H 6 ).25 Although the independence of critical shifts from the fluid type is verified in Figure 4, eqs 2 and 3 fail in describing PCP of fluids confined in SBA-15. The correlation of P Cp -shift, not T Cp -shift, is cubic as in eq 3 but with negative k 3 , which means that the shift dramatically increases when the


9


pores get smaller, consistent with the results of a molecular simulation of capillary condensation.26 Note that the correlation virtually coincides with eqs 2 and 3 only for large pores. On the other hand, the correlation of T Cp -shift is quartic, thus a new behavior. Therefore, the theoretical predictions need to be revisited, at least to deal with the experimental data for SBA-15. The correlations in Figure 4 are forced to pass the origin that represents the bulk CP:

σ  1− = q1 + q3    rp  PC rp   PCp

σ

3

2

(4a)

σ  σ  1− = q2   + q4    rp   rp  TC     TCp

4

(4b)

where the coefficients q 1 , q 2 , q 3 , and q 4 are 0.7689, 0.0519, 28.7529, and 25.7585, respectively, for fluids confined in SBA-15 in this study. CO2 (PCp) C2H6 (PCp) CO2 (TCp) C2H6 (TCp) Bulk CP Eq (2) Eq (3) Correlations

0.30

0.25

(1-TCp/TC) or (1-PCp/PC)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.20

0.15

0.10

0.05

0.00 0.00

0.05

0.10

0.15

σ/rp

0.20

0.25

Figure 4. The behavior of PCP-shift with respect to the ratio of molecular diameter to pore radius.


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In the process of locating PCP as described above, it is also possible to evaluate the behavior of heat of capillary condensation. The linearity shown in Figure 2 and eq 1 is consistent with a previous study that capillary-condensation pressures obey the Clapeyron equation as shown in Figure 5 for case S1 in the whole range of temperature:27

ln P = A + B / T

(5)

In SBA-15 (S1)

54.6

Pressure [bar]: natural logarithmic scale

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


46.0 37.3 CO2 28.7

C2H6

20.1 16.9 13.7 300

290

280

270

260

Temperature [K]: reciprocal scale

Figure 5. The validity of Clapeyron equation for CO 2 and C 2 H 6 in S1. Symbols are experimental data corresponding to those in Figure 2. For cases S2 and S3, see SI. where A and B are the regression coefficients. As v T ≈ v P , it can be shown that B ≈ b P /b T by substituting eqs 1a and 1b into eq 5. Further, B is the ratio of heat of capillary condensation LCC to the difference of compressibility factors ΔZ of the equilibrium phases:27

B=−

LCC / R  V PV  Z − Zl l   P  

=−

LCC / ΔZ R

(6)


11


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where the denominator of the first equation is the difference of compressibility factor of the equilibrium phases, i.e. the bulk vapor V and the confined liquid-like phase ℓ. The ratio of pressures PV/Pℓ is needed to account for the fact that the pressures of V and ℓ are different by the capillary pressure.28 Compressibility factor is defined as usual: Z=

P ρRT

(7)

where ρ is the phase density and R is the gas constant. The phase densities were not measured, so that we will present the ratio of LCC/ΔZ, which is calculable using eq 6. As the magnitude of Pℓ can be much larger than PV for fluids confined in nanopores,28,29 the difference of compressibility factors can be ΔZ ≈ ZV. For conditions where the bulk vapor behaves like ideal gas, ZV = 1, the ratio is effectively the heat of capillary condensation. However, we will not make any assumptions here, thus leave the ratio as it is. Any future work that can also measure the phase densities and their pressures would provide better estimate for the heat of capillary condensation. Tabel 3. Coefficients of the Clapeyron equation (eq 5), the ratio L′ = LCC/ΔZ, and the shift* ΔL′ = L′/ L′ bulk – 1; (for L′ bulk : LCC is replaced by the bulk heat of condensation) CO 2 SBA-15 A, B 11.352, S1 -2182.47 11.283, S2 -2156.91 11.237, S3 -2134.02

L′

C 2 H6 ΔL′

A, B

18.15±0.07 0.09337 kJ/mol

10.316,

17.93±0.04 0.08012 kJ/mol

10.256,

17.74±0.04 0.06868 kJ/mol

10.152,

-2014.44

-1989.20

-1946.09

L′

ΔL′

16.75±0.13 0.09836 kJ/mol 16.54±0.06 0.08459 kJ/mol 16.18±0.04 0.06098 kJ/mol

* The values in the bulk are roughly estimated by the midpoints of the range mentioned in the text: 16.60 kJ/mol for CO 2 and 15.25 kJ/mol for C 2 H 6 .


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The complete calculations for all cases are listed in Table 3. Calculated values of L′ = LCC/ΔZ are all higher than those of the bulk, i.e. ranging from 16.4 to 16.8 kJ/mol for CO 2 and from 14.9 to15.6 kJ/mol for C 2 H 6 , which can be derived from NIST data in the same range of temperature.27 As seen in Table 3, the ratio L′ systematically gets larger when the pore size is smaller. The shift of L′ from that in the bulk, i.e. L′/ L′ bulk – 1, increases quadratically as the pore size decreases as shown in Figure 6:

σ σ L′ − 1 = e1 − e2   rp ′ Lbulk rp 

   

2

(8)

where e 1 (= 0.829) and e 2 (= 1.596) are the regression coefficients.

0.10

CO2 C2H6 Correlation

0.08

L'/L'bulk - 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.06 0.04 0.02 0.00 0.00

0.04

0.08

0.12

0.16

0.20

σ/rp

Figure 6. The shift of the ratio L′ = LCC/ΔZ in SBA-15 from that in the bulk. The dotted curve is the correlation. The origin represents the bulk. 4. CONCLUSIONS AND REMARK


13


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In conclusion, the presence and location of PCP can be unambiguously demonstrated using our new isochoric cooling with DSC. While PCP is below CP for all cases, T Cp reduces from the bulk T C by an order of magnitude smaller than P Cp does from P C . T Cp and P Cp of fluids in SBA-15 increase with the pore size with different trends, both of which deviate from theoretical predictions. As a byproduct from this work, the total heat involved in capillary condensation was found to be linear both with 1/T and ln P in the whole range even close to PCP, which means that it does not follow the asymptotic power law near critical point as what happens in the bulk. Further, as Clapeyron equation is obeyed for all cases, the ratio LCC/ΔZ was found to become larger when the pore size is smaller. The findings of this study will allow not only more comprehensive investigation on PCP, but also better understanding of confined-fluid phase equilibria in general, including the heat involved in capillary condensation. Furthermore, revisiting theoretical studies on PCP is enabled due to future availability of experimental PCP data for any fluids confined in various nanopores that can be obtained using the technique introduced in this work. NOMENCLATURE A, B

correlation coefficients in eq (5)

b P , b T factors of slopes in eqs (1a) and (1b) c 1 , c 2 correlation coefficients in eq (2) e 1 , e 2 correlation coefficients in eq (8) h

total heat [J]

k 1 , k 2 correlation coefficients in eq (3) LCC

heat of capillary condensation [kJ/mol]

L'

LCC / ΔZ [kJ/mol]


14

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ΔL'

L'/ L' bulk - 1

P

pressure [bar]

PC

bulk critical pressure [bar]

P Cp

pore critical pressure [bar]

PV

pressure of vapor phase [bar]

Pℓ

pressure of confined liquid-like phase [bar]

ρ

fluid molar density [mol/m3]

q 1 -q 4 correlation coefficients in eqs (4a) and (4b) R

universal gas constant [J/(mol⋅K)]

rp

pore radius [nm]

σ

Lennard-Jones molecular diameter [nm]

T

absolute temperature [K]

TC

bulk critical temperature [K]

T Cp

pore critical temperature [K]

v P , v T factors of slopes in eqs (1a) and (1b) w

weight of adsorbent [mg], Table 2

Z

compressibility factor, eq (7)

ΔZ

difference of compressibility factor of equilibrium phases

ASSOCIATED CONTENT Supporting Information The following files are available free of charge. Experimental data and correlations (PDF)


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AUTHOR INFORMATION Corresponding Author * [email protected] Notes The authors declare no competing financial interests. ACKNOWLEDGMENT ST thanks the support from the Planetary Science Institute with the 2018 Incentive Awards. MD is grateful for the financial support from the Department of Petroleum Engineering in the College of Engineering and Applied Science at the University of Wyoming. ABBREVIATIONS DSC - Differential Scanning Calorimeter; PCP - pore critical point; HCP - hysteresis critical point; CP - critical point. REFERENCES (1) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; Sliwinska-Bartkowiak, M. Phase separation in confined systems. Rep. Prog. Phys. 1999, 62, 1573-659. (2) Barsotti, E.; Tan, S. P.; Saraji, S.; Piri, M.; Chen, J. H. A review on capillary condensation in nanoporous media: Implications for hydrocarbon recovery from tight reservoirs. Fuel 2016, 184, 344-361. (3) Evans R.; Marconi U. M.; Tarazona, P. Capillary condensation and adsorption in cylindrical and slit-like pores. J. Chem. Soc., Faraday Trans. 2 1986, 82, 1763-1787.


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TOC Image 4.7

Heat flow [mW]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 20

supercritical

4.6

subcritical 4.5

peak

4.4 304

300

296

292

288

Temperature [K]


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Critical Point of Fluid Confined in Nanopores: Experimental Detection

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